5 UPPER AND LOWER INTEGRALS 115 defined, and is denoted by £/„; also it belongs to £ (12.7.6). If we apply (13.5.2) to the sequence of partial sums ^fn9 and (13.5.3) to each of the »= i terms in this partial sum, we obtain the following corollary: (13.5.4) If(fn)nzi w & sequence of functions ^0 belonging to «/, then (00 \ 00 !/„ = lA/.). „=*! / „=! Now consider an arbitrary mapping / of X into R. There always exist functions heJ^ such that /z^/, for example the constant function equal to +00. Put (13.5.5) M*(/) = inf )u*(fc); this number £**(/) is called the upper integral of / with respect to the measure ju. If /e «/ it is clear that this definition agrees with the preceding one. Here the value of /**(/) can be any element of R. The relation /^ g implies /**(/) ^ M*(#)- F°r anY scalar a > 0 we have ti*(af) = <3ju*(/). (1 3.5.6) If the sumft +/2 o/ /wo mappings ofX into R /51 defined at all points o/X, and if n*^) > — oo and /J,*(f2) > — oo, This is obvious if one of the numbers n*(fi)> /**(/2) 1S +°°* If not, given any a > /^*(/!) and b > ju*(/2), there exist hl9 h2 in */ such that A ^ /zi, /2 ^ ^2 and ^(/zj) ^ a, ju*(/z2) ^ *• It follows that A4 + A2 ^/i +/2 and /X*(/ZA + /z2) ^ a + £ by (13.5.3). Hence the result. (13.5.7) If(fn) is any increasing sequence of mappings o/X into E such that > — oo for all sufficiently large n, then M*(sup/n\ \ n / = sup /**(/„) = lim The inequality ju*/sup/w\ ^ sup u*(fn) is clear. Let us prove the reverse inequality. We may assume that sup ju*(/M) < +00, otherwise there is n nothing to prove. By hypothesis, we may therefore assume that sup /x*(/n) n and all the ju*(/n) art finite. For each e > 0, we shall show that there exists an> the